ISSN:
1572-929X
Keywords:
43A65
;
22E45
;
35B45
;
35J15
;
35J30
;
58G03
;
35H05
;
22E25
;
Elliptic operators
;
hypoellipticity
;
regularity
;
semigroup kernels
;
kernel bounds
;
free nilpotent groups
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let (χ, G, U) be a continuous representation of a Lie groupG by bounded operatorsg →U(g) on the Banach space χ and let (χ, $$\mathfrak{g}$$ ,dU) denote the representation of the Lie algebra $$\mathfrak{g}$$ obtained by differentiation. Ifa 1, ...,a d′ is a Lie algebra basis of $$\mathfrak{g}$$ ,A i=dU(a i) and $$A^\alpha = A_{i_1 } ...A_{i_k } $$ whenever α=(i 1, ...,i k) we consider the operators $$H = \mathop \sum \limits_{\alpha ;|\alpha | \leqslant 2n} c_\alpha A^\alpha $$ where thec α are complex coefficients satisfying a subcoercivity condition. This condition is such that the class of operators considered encompasses all the standard second-order subelliptic operators with real coefficients, all operators of the form $$\sum _{i = 1}^{d'} \lambda _i ( - A_i^2 )^n $$ with Re λ i 〉0 together with operators of the form $$H = ( - 1)^n \mathop \sum \limits_{\alpha ;|\alpha | = n} \mathop \sum \limits_{\beta ;|\beta | = n} c_{\alpha ,\beta } A^{\alpha _* } A^\beta $$ where α*=(i k, ...,i 1) if α=(i 1, ...,i k) and the real part of the matrix (c α β) is strictly positive. In case the Lie algebra $$\mathfrak{g}$$ is free of stepr, wherer is the rank of the algebraic basisa 1, ...,a d′,G is connected andU is the left regular representation inG we prove that the closure $$\overline H $$ ofH generates a holomorphic semigroupS. Moreover, the semigroupS has a smooth kernel and we derive bounds on the kernel and all its derivatives. This will be a key ingredient for the paper [13] in which the above results will be extended to general groups and representations.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01468248
Permalink
